This invention relates generally to a novel device having applications in a number and variety of technological areas including recording devices and the like, i.e., a tapered resistive element employed in combination with selected heat sensitive media and more specifically an improved tapered resistive element which effectively reduces the power consumption in the specific application employed.
In accordance with the general concept of the instant invention, a device comprising a tapered resistance element which develops a non-uniform temperature profile on electrical energization is interacted after being energized with selected heat sensitive media to provide a number of very useful effects and devices which may be employed in a great many applications with ease, simplicity and greater economy than heretofore possible. For example, the intrinsic simplicity and low cost of this device opens new areas of applicability for analog measuring equipment as well as in existing application areas. The specifics of this novel device and its many and varied applications are more specifically outlined in U.S. patent application Ser. No. 747,167 filed concurrently herewith which is hereby respectfully incorporated by reference.
Generally described therein, a conventional resistive device is seen to be a resistive film having a uniform thickness which has been formed into a resistor of a specified width and length. This film is then placed on an insulating substrate which is bonded to a heat sink. When an electrical current I is passed through the resistor the production of Joule heat causes a steady state temperature above ambient .DELTA.T which if thermal fringing effects are neglected may be theoretically defined by the relationship .DELTA.T=(d.sub.s I.sup.2 .rho..sub.s)/(k.sub.s W.sup.2) (Eq. 1) in which d.sub.s and K.sub.s are respectively the thickness and thermal conductivity of the substrate and .rho..sub.s is the sheet resistivity of the resistive material measured in ohms/square. (Note: .rho..sub.s =.rho./d where .rho. is the bulk resistivity of the resistive material.) It is readily seen from this illustration that since the width of the resistor is uniform the local power dissipation and hence the temperature rise is also uniform so that no temperature gradient is established and the unique and utilizable effect of the device of the instant invention is not realized.
However, as is seen in FIG. 2 of the aforementioned U.S. Patent Application, a device may be provided including a resistive film which significantly has a varying width in the horizontal plane while the thickness remains uniform. This film may be placed on an insulating substrate which in turn is bonded to a heat sink. Now it is seen that the width of the resistive element is a monotonically increasing function of position along the length of the element or in simple terms the resistive element is tapered. In the event the slope of the taper is gradual over distances comparable with the substrate thickness equation 1 recited above will still be applicable for a first approximation. When a tapered resistor is energized the local power generation will vary along the length of the resistor so that points of prescribed temperature rise can be made to move along the tapered resistor by varying the current flowing through the device.
Although the non-uniformity of the width of the resistive film may vary in any suitable fashion, it is assumed for purposes of this discussion that the taper is linear so that the following relationship is theoretically true: w=w.sub.o +bx .theta.&lt;x&lt;1 (Eq. 2) in which w.sub.o is the width at the narrow end of the taper, b is the slope of the taper and x is the distance along the resistor measured from the narrow end. Assuming that the tapered resistive element is in contact with for example a thermographic substance which undergoes a color change when heated to the temperature T' or above as the current is increased in the tapered resistor a color line of x' will be drawn. The length of this line may theoretically be derived as follows: The temperature differential .DELTA.T' is defined as .DELTA.T'=T'-T.sub.amb' where T.sub.amb is the ambient temperature. Combining equations 1 and 2 yields the relationship between the applied current and the distance x' over which the tapered resistor will be heated to temperature T' or above, i.e., ##EQU1## It is seen that when w.sub.o is greater than zero no region of the taper will be hotter than T' for currents given by ##EQU2##
Typically, tapered resistors employing the system of the instant invention have been provided employing resistive metal film such as for example Ni.sub.80 Cr.sub.20 which has a constant resistivity over the temperature range of interest. For a gradual taper, the local temperature rise above ambient .DELTA.T is substantially given by .DELTA.T equals the quantity Ai.sup.2 .rho..sub.s all divided by W.sup.2 in which A is a constant, i is the current flowing in the tapered resistor, .rho..sub.s is the sheet resistivity of the resistor film and W is the taper width at the point of interest. In the event the given resistor employs a 3:1 taper (W.sub.max /W.sub.min =3) when the midpoint temperature rise is .DELTA.T.sub.0 the temperature rise at the narrow end is 4.DELTA.T.sub.o and at the wide end is 4.DELTA.T.sub.o /9. If .DELTA.T.sub.o represents the threshold temperature rise needed to create the desired thermographic indication then any regions heated significantly above .DELTA.T.sub.o results in unnecessary power dissipation as can be readily understood.
Thus it is an object of the present invention to provide an improved tapered resistor device.
It is another object of this invention to provide an improved tapered resistor device which reduces the power dissipation in regions of the resistor which are above threshold temperature.
Yet another object of this invention is to provide a tapered resistor device which reduces power consumption and yet provides sharpness of the temperature gradient at the region heated to the threshold temperature.